Suppose the number density of galaxies in the universe is, on average, \(3 \times 10^{-68}\) galaxy \(/ \mathrm{m}^{3}\). If astronomers could observe all galaxies out to a distance of \(10^{10}\) parsecs, how many galaxies would they find?

Understanding how to convert parsecs to meters is essential in astronomical calculations. A parsec (pc) is a unit of distance used in astronomy to describe vast spaces. One parsec is equivalent to approximately 3.085677581 × 10^{16} meters. This conversion factor is critical when working with distances in astronomical exercises. In our exercise, the distance given is 10^{10} parsecs. To convert this to meters, you simply multiply the number of parsecs by the conversion factor:

\[ d = 10^{10} \text{ parsecs} \times 3.085677581 \times 10^{16} \text{ meters/parsec} = 3.085677581 \times 10^{26} \text{ meters} \]

This makes it possible to work with distances in meters, which is a more standardized measurement in scientific calculations.

The volume of a sphere is calculated using the formula \[ V = \frac{4}{3} \times \pi \times r^3 \]. Here, \(\text{V}\) is the volume, \(\text{r}\) is the radius, and \(\text{pi}\) (π) is a constant approximately equal to 3.14159. This formula is essential when calculating the volume of spherical regions, such as parts of the universe. In our exercise, we converted the distance to meters, resulting in a radius of \[ r = 3.085677581 \times 10^{26} \text{ meters} \]. Plugging this into the volume formula gives:

\[ V = \frac{4}{3} \times \pi \times (3.085677581 \times 10^{26})^3 \approx \frac{4}{3} \times \pi \times 2.94 \times 10^{79} \approx 1.23 \times 10^{80} \text{ m}^3 \]

This step calculates the space in cubic meters, allowing us to determine how many galaxies could fit within that volume.

Astronomical calculations often involve determining quantities like the number of galaxies within a certain volume. For this, knowing the number density of galaxies is key. In our exercise, the number density of galaxies is given as \(3 \times 10^{-68} \text{ galaxies/m}^3\). By multiplying this number density by the calculated volume of the observable universe, you can find the total number of galaxies:

\[ N = 3 \times 10^{-68} \text{ galaxies/m}^3 \times 1.23 \times 10^{80} \text{ m}^3 \approx 3.69 \times 10^{12} \text{ galaxies} \]

This basic principle allows astronomers to estimate how many galaxies can be observed within a given distance. Such calculations are fundamental to understanding the scale and structure of our universe.